Bensa, Julien

Bilbao, Stefan

Kronland Martinet, Richard

Smith, Julius

English

Edinburgh Research Explorer

     Edinburgh Research Explorer                                      From the physics of piano strings to digital waveguidesCitation for published version:Bilbao, S, Bensa, J, Kronland Martinet, R & Smith, J 2002, 'From the physics of piano strings to digitalwaveguides'. in Proceedings of the International Computer Music Conference. pp. 45-48.Link:Link to publication record in Edinburgh Research ExplorerDocument Version:Preprint (usually an early version)Published In:Proceedings of the International Computer Music ConferencePublisher Rights Statement:© Bilbao, S., Bensa, J., Kronland Martinet, R., & Smith, J. (2002). From the physics of piano strings to digitalwaveguides. In Proceedings of the International Computer Music Conference. (pp. 45-48).General rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately andinvestigate your claim.Download date: 20. Feb. 2015From the Physics of Piano Strings to Digital WaveguidesJulien Bensa, Stefan Bilbao, Richard Kronland-Martinet and Julius O. Smith IIICenter for Computer Research in Music and Acoustics (CCRMA)Stanford Universityemail:bilbao@ccrma.stanford.eduAbstractA new model for transverse piano string vibration, second-order in time, which models frequency-dependent lossand dispersion effects is presented here. It is then shownhow a digital waveguide structure may be related di-rectly to this model. Finally, the model parameters arefit to experimental data from a grand piano.1 IntroductionSeveral models of transverse wave propagation ona piano string have appeared in the literature (Chaigneand Askenfelt 1994a; Chaigne and Askenfelt 1994b;Chaigne 1992). These models are always framed interms of a partial differential equation (PDE); usually,the starting point for such a model is the wave equation(Fletcher and Rossing 1991), and more realistic fea-tures, such as dispersion and frequency-dependent lossare incorporated through several perturbation terms. Themost advanced such model (Chaigne and Askenfelt 1994a)has been used as the basis for a high-fidelity sound syn-thesis technique (Chaigne and Askenfelt 1994b).Digital waveguides (Smith 1987; Karjalainen, Va¨lima¨ki,and Tolonen 1998) are filter-like structures which modelwave propagation as purely lossless throughout the lengthof the string, with loss and dispersion lumped in termi-nating filters. They are thus simulations of modifiedphysical systems, but are very efficient in the contextof musical sound synthesis. The aim of this paper is tolink PDE models and digital waveguides, and to explic-itly show the relationship between the lumped filtersused to model loss and dispersion and the parameterswhich define our PDE, which is a variant of Chaigneand Askenfelt’s system. Calibration of the filters tomeasured data is also discussed.2 PDE model of a stiff string, withfrequency-dependent lossIn this section, we present a new model of pianostring vibration, which can be written as2yt2= 22yx2  24yx4  2b1yt+ 2b23yx2t(1)Normally affiliated with S2M-LMA-CNRS, Marseille, France.Here, y(x; t) is the transverse displacement of the stringin a single plane, as a function of time t  0 and posi-tion x 2 [0; L℄ where L is the string length. The firstterm on the right-hand side of the equation, in the ab-sence of the others, gives rise to wave like motion, withspeed . The second “ideal bar” term introduces dis-persion, or frequency-dependent wave velocity, and isparametrized by a stiffness coefficient . The third andfourth terms allow for loss, and if b26= 0, decay rateswill be frequency-dependent. A complete model is ob-tained by including a hammer excitation term f(x; t),possibly including nonlinear effects, on the right-handside, and supplying a realistic set of initial and bound-ary conditions.If the term 2b23yx2tis replaced by 2b33yt3, we thenarrive at the Chaigne and Askenfelt model, and thusthe distinction is solely in the modeling of frequency-dependent loss. In a subsequent paper, we will describein detail the reasons why the model presented here ispreferable, but we briefly summarize them here: First,the restriction to second order of the time derivativesgreatly simplifies analysis and allows the constructionof more efficient finite difference schemes for whichnumerical stability is easily verifiable. Second, becausethe number of independent wave-like solutions is re-duced to two, it becomes possible to identify this equa-tion directly with digital waveguide models which arebased on the use of bidirectional delay lines which prop-agate traveling waves in opposite directions; for theChaigne model, third order in time, there are three in-dependent solutions. Finally, for a second-order equa-tion such as (1), generalization to more accurate mod-els of dispersion and loss (through the addition of moreterms to the PDE) can be achieved without disruptingthe well-posedness of the system (Strikwerda 1989),provided certain very simple conditions are respected.We note that in a later publication (Chaigne and Doutaut1997), a second-order model similar to the above waspresented in the context of finite difference schemes forxylophone bar vibration.It is also important to mention that the model (1),though convenient in that it allows very simple con-trol over the loss and dispersive characteristics of wavepropagation, is not completely satisfying when appliedto model real piano tones, which are, for much of therange of the piano keyboard, the result of two or threestrings struck simultaneously. Even as a representationof a single string, the term which models frequency-dependent loss does not, as far as we know, have asolid physical justification. The Chaigne and Asken-felt model, though, suffers from the same lack of phys-ical underpinnings, and we will take the same attitudeas these authors by treating it as a model which sum-marizes various physical processes (in particular loss),and which may be usefully be applied to the analysisand synthesis of musical tones.3 An associated waveguide model3.1 Dispersion and lossFor the purpose of examining the dispersion andloss characteristics of the traveling-wave solutions, theassumption of a string of infinite length is permissibleand simplifies the analysis; boundary conditions maybe reintroduced subsequently. Because the PDE (1) islinear and shift-invariant with respect to both time andspace, it can be analyzed by considering wave-like so-lutions of the formy(x; t) = est+jx (2)Here,  is the real spatial wavenumber, and s is a com-plex frequency variable. If such a solution is insertedin (1), then a dispersion relation (Elmore and Heald1969) results:s2+ 2(b1+ b22)s+ 22+ 22= 0 (3)(The analysis of the propagation characteristics of asingle wave-like solution can be considered to be a short-cut to the full Fourier and Laplace analysis, which achievesthe same resulting equation.) The roots of this equa-tion, s+and s , are thuss=  b1  b22p(b1+ b22)2  22  24(4)Over the range of  for which the quantity under theradical is negative (for b1and b2small, this will betrue for a substantial range of wavenumbers), sforma complex conjugate pair, and it is then possible to sep-arate the real and imaginary parts in terms of  by writ-ing s=   j!, with =  b1  b22 (5a)! =p22+ 24  (b1+ b22)2 (5b) is to be interpreted as a frequency-dependent loss pa-rameter (notice that for b1and b2chosen non-negative,we have   0 for all , so exponential solutions arealways damped), and ! is a real frequency variable.3.2 The corresponding waveguide filterWe now turn to the problem of relating digital waveg-uide parameters (to be discussed shortly) to the loss andinharmonicity parameters discussed in Section 3.1. Tothis end, we now return to the string of length L, underfixed boundary conditions. A simplified digital waveg-uide structure which can be used for piano synthesis isas shown in Figure 1.D0FOIFigure 1: Waveguide filter structure.The excitation I corresponds to the energy suppliedto the string initially at one endpoint (by the hammer).The output O will then represent the time waveform ofthe string’s motion at the bridge, or perhaps a unidi-rectional traveling-wave component at a selected pointalong the string. We will denote by D the time takenfor energy in the lowest mode to complete a round-trippassage of distance 2L over the string; this minimumdelay is simulated by a digital delay line of durationD0(as shown in Figure 1). The lumped digital filter Fsimulates the remaining accumulated lossy and disper-sive effects over the same round-trip propagation dis-tance; in particular it accounts for losses in the string it-self (from air friction, and viscoelastic effects), as wellas losses at the endpoints (where energy is transmittedinto the soundboard).Consider an exponential wave solution propagatingalong the string, over a time duration D. From the def-inition of the exponential solution from (2), the prop-agation can be represented by a multiplicative phasefactor exp(sD + 2jL). The modulus and phase ofthis factor are thus related to the filter F byjF j = eD= e b1D b2D2 (6)arg(F ) = !D   2L (7)Thus the phase of F is that of the multiplicative phasefactor, except for the removal of the constant delayIn order to rewrite this filter in terms of the fre-quency !, it is necessary to express the wavenumber in terms of !. From (5b), and solving for , the re-quired expression will be = s q pq2+ 4r(b21+ !2)2r(8)withq = 2  2b1b2r = 2  b22(9)Given that, for realistic piano string modeling, b1' 1and b2' 10 4, we make the simplifying assumptionsb1b2 2b22 2b21 !2 (10)which permit the following approximation of : ' r222(11)where =p1 + 42!2=4  1 (12)Finally, we arrive at approximate expressions for themodulus and phase of the filter F as a function of fre-quency !,jF j ' exp Db1+b2222(13a)arg(F ) ' !D   Lp2(13b)4 Calibration of loss and inharmonic-ity parameters from experimen-tal dataBecause this model is intended for use in the con-text of musical sound synthesis, we here discuss thecalibration of b1and b2, and the determination of thestiffness parameter. To this end, a large number ofmeasurements were taken, using a Yamaha Disklavier;for each note, the vibration of the string at the bridgewas measured (using an accelerometer), for a hammerspeed of 2.1 m/s.Using signal processing techniques (Aramaki, Bensa,Daudet, Guillemain, and Kronland-Martinet 2002), thedamping coefficients were then determined. Over mostof the piano range, however, the hammer strikes notone, but two or three strings simultaneously. The cou-pling gives rise to perceptually significant phenomenasuch as beating and two-stage decay; these effects arenot taken into account in model (1). For these multi-string notes, the calculated damping coefficients can bethought of describing the global perceived decay of thesound. For each note, b1and b2were calculated from(13a). The evolution of these parameters as a functionof MIDI note value is shown in Figure 2.b1and b2are both increasing functions of MIDInote value, indicating increasing loss as one approachesthe treble range. The physical characteristics of stringsthemselves, however, vary only slightly for wrappedstrings, and thus the variations in the loss parameterswould appear to be due to boundary termination. Inthe simple model we have presented, boundary condi-tions were assumed to be lossless, but in a real piano,the loss is extremely important, as it is the mechanismby which energy is transferred to the soundboard, and,ultimately to the listener as a musical sound. Theselosses will be greater in the treble range than in thebass, because strings are shorter, and waves are ableto complete more round-trip passages in a given time.Thus, although our model does not fully describe stringvibration in a true piano, we have calculated “equiva-lent” parameters b1and b2.30 40 50 60 70123451/sb130 40 50 60 701234567x 10−4note (MIDI)m2 /sb2Figure 2: Values of b1and b2fitted from measured dataas a function of MIDI note value.In Figure 3, we have also fit simple curves to theloss parameter data. The fits are linear as a function ofthe fundamental frequency, and are given byb1= 7:4 10 3f0  4:49 10 2 (14a)b2= 1:0 10 6f0+ 1:07 10 5 (14b)These simple empirical descriptions of b1and b2al-lows the reproduction of piano tones whose dampingwill be identical to that of the perceived acoustic note.The study of more detailed models of multi-string cou-pling is currently in progress; a multi-string waveg-uide model was presented in (Aramaki, Bensa, Daudet,Guillemain, and Kronland-Martinet 2002).In practice, it is helpful to work with more per-ceptually significant parameters. We may express thephase of the filter F in terms of the inharmonicity co-efficient B (H. Fletcher and Stratton 1962) and the fun-damental frequency !0byB = 2!20=4 = !0L= (15)from which, the phase of F may be rewritten asarg(F ) ' !D  p2pBr 1 +q1 + 4B!2=!20(16)This expression for the phase allows the estimationof !0and B for each note. The inharmonicity factorB is plotted as a function of MIDI note value in Figure3; B is an increasing function of the note value, ex-cept over the bass range, where the strings are double-wrapped.The determination of b1, b2, B and !0for each noteallows us, then, an explicit expression of the behaviorof the filter F as a function of note value. In order20 30 40 50 60 7000.20.40.60.811.2x 10−3 BnoteFigure 3: The measured inharmonicity factor B.to represent the evolution of the loop filter in terms ofnotes, we show in Figure 4 the modulus and phase ofthe elementary filter ÆF , normalized with respect to thefilter D:ÆF = F1=D (17)The modulus, which also takes into account the lossesat the endpoints, is decreasing with the note. But wecan notice that this behavior is slight different for thewrapped strings (A0, A1) than for the other strings (A2,E3, A3). It is worth noting that although B is an in-creasing function of note value, the phase of the fil-ters ÆF grows less rapidly. This is due to the fact thatthe phase of the filter depends not only on B, but alsothe fundamental frequency. This effect can be under-stood by expanding the expression for the phase of Ffor 4B(!=!0) near zero; one getsarg(F ) ' !2B2!20(18)Though meaningful only for the first few partials, onecan clearly see the dependence which is decreasing withthe fundamental frequency.0 2000 4000 6000 8000 100000.99970.99980.99991normalized modulus of the filterA0A1 A2E3A30 2000 4000 6000 8000 100000123x 104 normalized phase of the filterfrequency (Hz)A0A1 A2E3A3Figure 4: Normalized modulus and phase of F for se-lected pitches.5 ConclusionsWe have presented in this paper a new model of pi-ano string vibration, which takes into account effectsof stiffness and frequency-dependent loss. This modelcan be considered to be a variant of the model of Chaigneand Askenfelt which possesses the advantage that ex-plicit solutions may be derived and related to a digi-tal waveguide structure, to be used for musical soundsynthesis purposes. We also have shown how the pa-rameters which define the model may be calibrated toexperimental data and have provided a simple descrip-tion of the variation of these parameters of the musicalrange of the piano.Future work will involve extending the identifica-tion with a PDE to the multi-string framework.ReferencesAramaki, M., J. Bensa, L. Daudet, P. Guillemain, andR. Kronland-Martinet (2002). Resynthesis of coupledpiano string vibrations based on physical modeling.Journal of New Music Research 30(3), 213–26.Chaigne, A. (1992). On the use of finite differences formusical synthesis. Application to plucked stringed in-struments. J. d’Acoust. 5(2), 181–211.Chaigne, A. and A. Askenfelt (1994a, Feb.). Numeri-cal simulations of struck strings. I. 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Music applications of digital waveguides.

Numerical simulations of struck strings. I. A physical model for a struck string using ﬁnite difference methods.

Numerical simulations of struck strings. I. Comparisons with measurements and systematic exploration of some hammer-string parameters.

Numerical simulations of xylophones. II. Time-domain modeling of the resonator and of the radiated sound pressure.

On the use of ﬁnite differences for musical synthesis. Application to plucked stringed instruments.

Physics of Waves.

Quality of piano tones.

Resynthesis of coupled piano string vibrations based on physical modeling.

The Physics of Musical Instruments.

From the physics of piano strings to digital waveguides

Julien Bensa

Stefan Bilbao

Richard Kronland-Martinet

Julius O. Smith III

University of Michigan Library Repository

From the Physics of Piano Strings to Digital Waveguides

http://hdl.handle.net/2027/spo.bbp2372.2002.009

From the physics of piano strings to digital waveguides

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